X-ray study of the second-order phase transition of Ag 0.35TiS 2: A phase transition characterized by two order parameters

Abstract

Ag 0.35TiS 2 shows a second-order phase transition with T c = 298.0 K from a a√3 × a√3 × 2 c superstructure, space group P 31c , to a disordered Cd(OH) 2NiAs intermediate structure with unit cell a × a × c and space group P 3m1 . In the disordered structure silver atoms occupy fractionally the sites of triangular lattice planes. Below T c silver atoms occupy special sites in space group P 31c : Ag(α) at ±(0, 0, 1 4 ) , Ag(β) at ±( 2 3 , 1 3 , 1 4 ) , and Ag(γ) at ±( 1 3 , 2 3 , 1 4 ) . The order is characterized by two order parameters η 1 = n( β) − n( γ) and η 2 = n( β) + n( γ) − 2 n( α), where n( α), n( β), and n( γ) are the occupancies of the α, β, and γ sites, respectively. Single-crystal X-ray diffraction of the ordered structure showed the occupancies n( α) = 0.172, n( β) = 0.861, n( γ) = 0 at 100 K and n( α) = 0.180, n( β) = 0.786, n( γ) = 0.069 at 255 K. Critical exponents β 1 and β 2 of the order parameters η 1 and η 2 were determined from the temperature dependence of the intensities of the superreflections 101 and 102, respectively; the values obtained, β 1 = 0.41 and β 2 = 0.81, are between those predicted by P. Bak ( Phys. Rev. Lett. 44, 889 (1980)) from scaling laws: β 1 = 0.33 and β 2 = 0.86, and from mean field theory: β 1 = 0.5, β 2 = 1.0. The temperature dependence of the order parameters in the temperature range 100–298 K is compared with that from a Bragg-Williams (mean field) model, the energy being written in terms of nearest-neighbor repulsive interactions, ε 1 for atoms at distance a and ε 2 for atoms at distance c; ε 1 > ε 2 because c > a. The space group of the observed structure, P 31c , is discussed in (3 + 1)D superspace, which gives an alternate description for the symmetry of this commensurately modulated phase. Finally, some related phases are discussed.